L(s) = 1 | − 2-s + 4-s + 5-s − 1.60·7-s − 8-s − 10-s − 3·11-s − 0.167·13-s + 1.60·14-s + 16-s + 2.93·17-s + 5.20·19-s + 20-s + 3·22-s − 7.20·23-s + 25-s + 0.167·26-s − 1.60·28-s − 2.16·29-s + 5.03·31-s − 32-s − 2.93·34-s − 1.60·35-s − 4.03·37-s − 5.20·38-s − 40-s − 6.93·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.605·7-s − 0.353·8-s − 0.316·10-s − 0.904·11-s − 0.0463·13-s + 0.428·14-s + 0.250·16-s + 0.712·17-s + 1.19·19-s + 0.223·20-s + 0.639·22-s − 1.50·23-s + 0.200·25-s + 0.0327·26-s − 0.302·28-s − 0.402·29-s + 0.904·31-s − 0.176·32-s − 0.503·34-s − 0.270·35-s − 0.663·37-s − 0.844·38-s − 0.158·40-s − 1.08·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
good | 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 0.167T + 13T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 - 5.20T + 19T^{2} \) |
| 23 | \( 1 + 7.20T + 23T^{2} \) |
| 29 | \( 1 + 2.16T + 29T^{2} \) |
| 31 | \( 1 - 5.03T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 + 6.93T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 0.501T + 47T^{2} \) |
| 53 | \( 1 - 0.167T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 - 4.03T + 61T^{2} \) |
| 71 | \( 1 + 2.83T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 - 0.334T + 79T^{2} \) |
| 83 | \( 1 + 4.63T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 1.79T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78140255529936898981375696635, −7.16926377138458813031963407858, −6.32980168474098523290657127874, −5.68183533876186855956106487411, −5.07667970212848676728704439173, −3.85329013368487448551681385275, −3.03172346985942449672839256292, −2.29130845054801435455649016508, −1.22536885269975029569105231222, 0,
1.22536885269975029569105231222, 2.29130845054801435455649016508, 3.03172346985942449672839256292, 3.85329013368487448551681385275, 5.07667970212848676728704439173, 5.68183533876186855956106487411, 6.32980168474098523290657127874, 7.16926377138458813031963407858, 7.78140255529936898981375696635